Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = 26x^6 - 43x^5 - 60x^4 + 86x^3 + 62x^2 - 47x - 30$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = 26x^6 - 43x^5z - 60x^4z^2 + 86x^3z^3 + 62x^2z^4 - 47xz^5 - 30z^6$ | (dehomogenize, simplify) |
$y^2 = 105x^6 - 172x^5 - 238x^4 + 344x^3 + 249x^2 - 188x - 120$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-30, -47, 62, 86, -60, -43, 26]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-30, -47, 62, 86, -60, -43, 26], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-120, -188, 249, 344, -238, -172, 105]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(62400\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5^{2} \cdot 13 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(62400\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5^{2} \cdot 13 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(746956\) | \(=\) | \( 2^{2} \cdot 7^{2} \cdot 37 \cdot 103 \) |
\( I_4 \) | \(=\) | \(964099\) | \(=\) | \( 967 \cdot 997 \) |
\( I_6 \) | \(=\) | \(239731805892\) | \(=\) | \( 2^{2} \cdot 3 \cdot 19 \cdot 1051455289 \) |
\( I_{10} \) | \(=\) | \(7800\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \) |
\( J_2 \) | \(=\) | \(746956\) | \(=\) | \( 2^{2} \cdot 7^{2} \cdot 37 \cdot 103 \) |
\( J_4 \) | \(=\) | \(23246993348\) | \(=\) | \( 2^{2} \cdot 53 \cdot 109655629 \) |
\( J_6 \) | \(=\) | \(964640334026944\) | \(=\) | \( 2^{6} \cdot 23 \cdot 2713 \cdot 8663 \cdot 27883 \) |
\( J_8 \) | \(=\) | \(45030296405368433340\) | \(=\) | \( 2^{2} \cdot 3^{5} \cdot 5 \cdot 83 \cdot 10349707 \cdot 10786049 \) |
\( J_{10} \) | \(=\) | \(62400\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5^{2} \cdot 13 \) |
\( g_1 \) | \(=\) | \(3633248698565865263943963184/975\) | ||
\( g_2 \) | \(=\) | \(151381177054798833726360812/975\) | ||
\( g_3 \) | \(=\) | \(8409602787821673218459056/975\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
No rational points are known for this curve.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable everywhere.
magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.459086\) | \(\infty\) |
\(D_0 - D_\infty\) | \(7x^2 - 4xz - 8z^2\) | \(=\) | \(0,\) | \(98y\) | \(=\) | \(-121xz^2 - 32z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(5x^2 - 2xz - 5z^2\) | \(=\) | \(0,\) | \(25y\) | \(=\) | \(-27xz^2 - 5z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.459086\) | \(\infty\) |
\(D_0 - D_\infty\) | \(7x^2 - 4xz - 8z^2\) | \(=\) | \(0,\) | \(98y\) | \(=\) | \(-121xz^2 - 32z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(5x^2 - 2xz - 5z^2\) | \(=\) | \(0,\) | \(25y\) | \(=\) | \(-27xz^2 - 5z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - xz^2\) | \(0.459086\) | \(\infty\) |
\(D_0 - D_\infty\) | \(7x^2 - 4xz - 8z^2\) | \(=\) | \(0,\) | \(98y\) | \(=\) | \(x^3 - 241xz^2 - 64z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(5x^2 - 2xz - 5z^2\) | \(=\) | \(0,\) | \(25y\) | \(=\) | \(x^3 - 53xz^2 - 10z^3\) | \(0\) | \(2\) |
2-torsion field: 8.8.94758543360000.5
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 0.459086 \) |
Real period: | \( 12.07414 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.385769 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(6\) | \(1\) | \(1 + T\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) | |
\(5\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 6 T + 13 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.90.6 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);